RELATIONSHIP BETWEEN ZEROS AND COEFFICIENTS OF A POLYNOMIAL WORKSHEET

Problem 1 :

If α and β are zeros of p(x) = x² + x - 1, then find 1/α+ 1/β?

Solution

Problem 2 :

If one of the zeros of the cubic polynomial x³ + ax² + bx + c is -1, then what will be the product of the other two zeros?

Solution

Problem 3 :

If α, β, γ be zeros of the polynomial 6x³ + 3x² -5x+1, then find the value of α^-1 + β^-1 + γ^-1?             Solution

Problem 4 :

If α, β are zeros of the quadratic polynomial

f(x) = 2x² + 11x + 5

find

a) α⁴ + β⁴      b)1/α +1/β -2αβ

Solution

Problem 5 :

If the zeros of the polynomial f(x) = x³ – 3x² - 6x + 8 are of the form a-b, a, a+b, then find all the zeros

Solution

Problem 6 :

If α, β are the two zeros of the polynomial

f(y) = y² - 8y + a and α² + β² = 40

find the value of ‘a’?

Solution

Problem 1 :

The quadratic equation kx² + (k–8)x + (1 – k) = 0, k ≠ 0, has one root which is two more than the other. Find k and the two roots.

Solution

Problem 2 :

The roots of the equation x² – 6x + 7 = 0 are α and β. Find the simplest quadratic equation with roots

α + 1/β and β + 1/α.

Solution

Problem 3 :

The roots of 2x² – 3x - 5 = 0 are p and q. Find all quadratic equations with roots p² + q and q² + p.

Solution

Problem 4 :

kx² + (k + 2) x - 3 = 0

has roots which are real and positive. Find the possible values that k may have.

Solution

Problem 1 :

Which are the zeros of p(x) = x² + 3x - 10.

a)  5, -2       b)  -5, 2       c)  -5, -2       d)   none of these

Solution

Problem 2 :

Which are the zeros of p(x) = 6x² - 7x + 12.

a)  5, -2         b)  -5, 2       c)  -5, -2      d)  none of these

Solution

Problem 3 :

Which are the zeros of p(x) = x² + 7x + 12.

a)  4, -3        b)  -4, 3       c)  -4, -3     d)  none of these

Solution

Problem 4 :

If the product of zeros of the polynomial ax² - 6x - 6 is 4, find the value of ‘a’.

Solution

Problem 5 :

If one zero of the polynomial (a² + 9)x² + 13x + 6a is reciprocal of the other. Find the value of a.

Solution

Problem 6 :

Find the zeros of the quadratic polynomial x² + 5x + 6 and verify the relationship between the zeros and coefficients.

Solution

Problem 7 :

Find the zeros of the polynomial

p(x) = √2x² - 3x - 2√2.

Solution

Problem 8 :

If α, β are the zeros of the polynomials

f(x) = x² + 5x + 8

then α + β

a)  5       b)  -5       c)  8      d)  none of these

Solution

Problem 9 :

If α, β are the zeros of the polynomials

f(x) = x² + 5x + 8, then α ∙ β

a)  0      b)  1      c)  -1      d)  none of these

Solution

Problem 10 :

The value of k such that the quadratic polynomial

x² - (k + 6)x + 2(2k + 1)

has the sum of zeroes as half of their product is 

a)  2      b)  3        c)  -5      d)  5

Solution

Problem 1 :

Find zeroes of the quadratic polynomial

x² + x – 2 = 0

a. 4, -5       b. 2, -4       c. 2, -1      d.1, -2

Solution

Problem 2 :

Write a quadratic equation with the given roots. Write the equation in the form ax² + bx + c = 0, where a, b, and c are integers. 

-5 and -1

Solution

Problem 3 :

Write a quadratic polynomial, sum of whose zeroes is 2√3 and product is 5.

Solution

Problem 4 :

For what value of k, (–4) is a zero of the polynomial

x² – x – (2k + 2)?

Solution

Problem 5 :

For what value of p, (–4) is a zero of the polynomial

x² – 2x – (7p + 3)?

Solution

Problem 6 :

If 1 is a zero of the polynomial p(x) = ax² – 3(a – 1) x – 1, then find the value of a.

Solution

Problem 7 :

If (x + a) is a factor of 2x² + 2ax + 5x + 10 find a.

Solution

Problem 8 :

If one zero of the polynomial f(x) = (k² + 4) x² + 13x + 4k is the reciprocal of the other, then k is equal to 

a)  2         b)  -2         c)  1       d)  -1

Solution

Problem 9 :

If α and β are zeroes of the polynomial x² + 5x + c and α - β = 3, then find c.

a)  2         b)  3         c)  4       d)  1

Solution

Problem 10 :

For what value of p, 1 is a zero of the polynomial f(x) = 2x² + 5x - (3p + 1) ?

a)  3      b)  5        c)  2      d) -1

Solution